Fractals and Stories
A decoration for Contextmas 2012, requested of @vruba by Jessica Ferris.
Bref, le monde d’abord; puis, dans le monde, l’homme; dans l’homme, les autres hommes.
—Luc Durtain, D’homme à homme.
I’m going to ask you to think about something that may seem dull. It’s a trick. The closer you follow, the cooler it will be. Now watch my hands closely.
Take a grid, like graph paper, that’s labeled with an x and y axis. Draw a circle that’s 4 units wide, centered at x = 0, y = 0. Put a bead somewhere in the circle and mark its starting x and y. Now move it according to this formula:
- Find the square of its present x position, subtract the square of its y position, and add the original x position. Slide it to that value on the x axis.
- Double its present x times its present y, and add the original y. Move the bead to that point on the y axis.
Repeat. If the bead doesn’t leave the circle before you get bored, leave the dot where it started. If it does leave the circle, erase it. Then try with another starting point.
If we take the graph paper to be the complex plane (in the math sense), and the bead is named z and its starting point is c, there’s a neater way of writing this: z ⟼ z2 + c. The various arithmetic operations were mostly for the z2 – exponentiation of a complex number is finicky.
I’d like you to convince yourself that the bead game isn’t magic. There’s nothing tricky in the rules, just schoolbook addition, subtraction, and multiplication.
In this visualization the circle is purple, and overlaid on it in black is the zone where the bead won’t leave the circle on the first move of the game. Some areas in the black might take two or more moves for the bead to leave; on other starting points in there, it might end up in a never-ending loop:
I’m adding crazy fuchsia for the zone that the bead can leave the circle from in two moves, so black is for three or more:
Colors for up to four moves:
Looks kind of like a ripply manta, doesn’t it? Up to eight moves:
Up to 256 moves:
The bead game reveals the Mandelbrot fractal. More precisely, the Mandelbrot Set is the area where the bead would never leave the circle even given an infinite number of moves.
Let’s adjust the color so that successive moves blend more smoothly …
… and zoom into the cleft between the biggest bulbs:
Every part of the shore between the Mandelbrot Set and its surroundings has its own version of the universal bulb pattern. The regions have names like Elephant Valley, Double Scepter Valley, and Triple Spiral Valley.
We’re in Seahorse Valley.
And we could go on forever – as far as anyone knows. On YouTube you can find zooms that cover a range of scales equivalent to the ratio between the sun and an atom. There’s a whole microculture of sharing fun coordinates: I found this one (0.743 643 887 037 151 −0.131 825 904 205 330i) in a lovely series on Wikipedia by Wolfgang Beyer.
Beyond depth, through flavors
The long zooms are amusing to me, but only amusing. I can sit and watch for a long time, and they can induce a kind of mild contemplation, and then I find myself wondering what the point is.
A more interesting feature of the Mandelbrot fractal, for me, is the flavors. Each region brings its own accent or style to the repeating motifs. In Double Scepter Valley, the little bulbs all carry double scepters; in Seahorse Valley, they all sprout tails. There are no borders between styles. Between Double Scepter Valley and Seahorse Valley, the curlicues are intermediate. They must have discovered z ⟼ z2 + c elsewhere in our galaxy, and in other galaxies, and there our seahorses and double scepters might be the unnamed intermediates between Bumblecrow Wedge and Wind-Spun Fronds Wedge or Bufflegoat Wedge. It’s all quite fluid for something defined by a few lines of arithmetic.
And the most interesting thing to me is not even known. It’s an open question in mathematics: MLC, the Mandelbrot local connectivity conjecture. If it’s true, and so far there’s neither proof nor disproof, then the whole set, the entire black area, is all connected, with a single outline. No holes – and no islands. If we zoomed in far enough, we would find a tiny filament linking the lonely-looking Mandelbaby in Seahorse Valley with the big one.
And into branches
Which brings us to branches. Nature is full of branches because if you want to be close to a lot of space with minimal increase in your own volume, your best bet is a structure that forks and sub-forks. This is how birds make wings: they send out a bone with a little quick tissue and feather roots, and from them grow rachis, and from them grow barbs, and from them grow barbules, and from them grow barbicels, between which the gaps are so small that air doesn’t bother trying to get through, and the bird has made something the shape of an airfoil without having to fill it with heavy meat. Branching is also how plants fill out their space in the air to collect sunlight and CO2, and underground to find solid nutrients. The structures of your nervous, respiratory, and circulatory systems are similarly manifold.
But the principle is deeper than life itself. It’s how rivers work: though we don’t ascribe intent, a river
needs to approach every part of its basin in the sense that erosion will make it grow streams and rivulets until it does. It doesn’t even have to be water. Here’s an image search for
viscous fingering. What could go wrong? Fluid dynamicists use the phrase viscous fingering to describe this kind of structure in their domain.
In fact, apparently in some sense it might be deeper than physics, because – assuming MLC is true – the Mandelbrot, a piece of abstract math, has some desire-less
need to connect its infinite brood.
Before we leave the Mandelbrot behind, I think it might be worth your time to revisit it, knowing what you know now. Scroll up to the top and re-read the description of the bead game. Consider its simplicity. And as you scroll back to here, keep in mind that every image in the zoom flows entirely out of that game and a few choices of color and framing. Infinite seahorses are curled inside the idea of counting iterations of z2 + c. I’ll see you again in thirty seconds.
- It’s similar at every scale;
- it’s full of continuous variations; and, conjecturally,
- it’s all connected.
A highly realistic painting is a lot of work. Redoing it from a slightly different perspective with perfect duplication of tone and detail would be far more work – you would be painting both the scene and your first painting of it. When photography was new, it wasn’t all that much faster or cheaper than painting, but one of its great appeals was that making two photographs of something at once was only twice as much work as a single photograph. Early photography is full of stereoscopic images –
stereós being Ancient Greek for
When two-channel recording was invented, it seemed natural to call it
stereophonic, and when we started shortening that to
stereo, we lost the earlier sense of the morpheme. But it persists in another word:
stereotype. A sterotype was a block of type fused solid – a backtrack from movable type for the sake of convenience. A newspaper might have a stereotype for
the or the name of its town; Utilities like TextExpander and autocomplete serve a purpose very much like that of stereotypes. a jobbing printer might make a stereotype of a client’s business card so it would wear longer without requiring resetting. (There was another word for this kind of object, from French onomatopoeia: cliché.)
Types, in a related but different sense, were of particular interest to a lot of the people taking stereographic photos. When I searched the Library of Congress prints and photographs collection for example stereographs, I found a lot of things like this:
Judging by how worn and (as far as I can tell) ill-fitting her clothing is, I think it’s safe to say that she is very poor by my standards. But the clothing itself would be unaffordable to me, because it’s hand-embroidered. If you happen to surf the Library of Congress prints and photographs collection and look at the various kinds of distancing, exoticization, and simplification – stereotyping – in early photography, you will sooner or later notice that you keep seeing amazing decorative art, especially embroidery.
I could pull up dozens more pictures just on the Library of Congress site. I haven’t even touched on painting, nor the entire Southern Hemisphere.
This is a pre-modern look that has nothing to do with the quality of the film. Nor is it pre-modern – what I mean is non-industrial. It’s the branches. People who do not use mass production for their decoration often use roughly space-filling curves: patterns that can be elaborated until they cover their base fabric. It is neither necessary nor sufficient I encourage someone else to write their own speculative nonfiction about why this is. I suspect that William Morris, studies of time use, and a diagram of a loom will feature heavily. to be away from contemporary Western technology to embroider – or paint, or engrave – manifold branches on things, but the correlation seems strong.
It’s busy, interlocking patterns. It’s a line of decoration along every border. I don’t know exactly. (There’s a reason I tried to show it instead of tell it.) It’s been out of prestige fashion in the English-speaking world since, oh, maybe Hogarth. In the everyday culture around me, if a created visual thing is off-grid, peritactic, lacks heirarchical structure, is adjusted to fit the irregularities of its background, and doesn’t leave much of its area blank, it tends to be seen as cheap-looking at best. It may even be treated as the product of pathology. In fact, I think a lot of folk art in more or less mainstream American culture comes from things that it is not wrong to treat as mental illness – which is very different from saying that it is not art.
One of the things that art by obsessive people can help us with is figuring out where our conventions are. Fractals are like this as well, because they too are from the outside – although from a completely different angle. One thing they both teach me is that I find it very strange to be confronted with many variations of a motif at once. If you have a cross, I ask myself, why would you make another and put them together without any particular arrangement? What is the confusion of esthetic responses when I see how every seahorse tail is slightly different from the one next to it?
If no one had heard of the Odyssey, and you tried to publish it today as fiction set in the world of the Greek myths, your editor would probably praise many parts highly, then gently outline some concerns. For starters, it’s about as structured as a handful of sand. Theoretically, the hero is driven by his desire to get home, but this only affects him at random and rarely in interesting ways. For the most part, he’s having entertaining but arbitrary adventures among supporting characters we never get to know. There’s little
because … and so … but finally and a whole lot of
and then … and then … and then. It’s like a comic book series with good issues but no proper continuity or arc. It’s like a beautifully rendered version of something designed by a small child. And then of course the repetition: Odysseus is called by the same adjectives dozens of times. Entire metaphors appear twice. Some lines seem plagiarized from earlier chapters. Even a few relationships are similar in ways that don’t look deliberate.
In short, the Odyssey – with all its brilliance – is written like a crazy person writes. This caused a lot of problems for modern European scholars of literature, who were curiously unable to think of anything held in high regard by the Classical Greeks as imperfect in more than one or two details.
Over several decades, the scholars figured out how to avoid calling Homer crazy, but at an awful price.
Matija Murko, building on the work of Vuk Karadžić and others, had wandered around the Balkans in the first decades of the twentieth century and recorded living folktales. (There’s a terminological subtlety here – to spare certain sensitivities, when a folktale is told by an adult man, it’s conventionally referred to as an
epic. The stories Murko found were told by women as well as men, but were considered as primarily coming from men.) The areas with the richest remaining traditions tended to be those with the least state power and the lowest literacy, due to poverty or war. Murko was very explicit that the stories were not fixed things (from Singers and the Epic Songs, c. 1928, in Oral Tradition 5/1 (1990), translated by John Miles Foley – bracketed notes here are from that translation):
A singer can also modify songs as he goes, according to the time available, his own mood, the audience before whom he is performing, and the payment he has reason to expect. Moreover, the audience can directly influence him, and, when a song lasts too long, someone may cry out to him:Goni, goni! (“faster, faster!” [lit. “get going, get going!”]). I cite the example of a certain prisoner from Lepoglava in Croatia, from whose dictation songs of 2500 and 4400 verses were transcribed, even though the same songs had no more than 1200 and 1500 verses when sung by the man in Bosnia who had taught him.
Murko reports that he met one Salko Vojniković, who, over six weeks, sang about twice as many words as the Iliad and the Odyssey combined. He remarks
And there have been philologists who doubted that a single singer could know all of Homer by heart!, and, a little later:
It is absolutely certain that under such conditions a text cannot remain unchanged. I have demonstrated this experimentally. On two occasions I brought with me a phonographic apparatus perfected by the Viennese Academy. I could not record the long epic songs on this machine, but it did suffice for fragments of less than 30 verses to verify something unexpected. Since it was necessary to write down each text before phonographic recording, I asked the singer first of all to practice outside the tent while a stenographer transcribed the text. I thus had three texts at the same time from a single session, and even four in one case. The comparison showed that not only isolated words or word-order but entire verses appeared in a wholly new form or simply disappeared, so that of 15 dictated lines [in one version], for example, there might remain [in the next version] no more than 8 sung lines. A fine singer from northwest Bosnia himself modified the opening line on each occasion.
He said the first time:
Beg Osman beg rano podranioAn etymological figure is the repetition of a word (in some form) as a different part of speech within a single phrase:she rode the rideorthe rain rained. Here it would literally be something likeOsmanbeg early-rose early.(etymological figure)
“Osmanbeg arose early”;
then while practicing:
Beg Osman beg na bedem izidje.
“Osmanbeg mounted the ramparts”;
and afterward he sang:
Beg Osman beg niz Posavlje gleda.
“Osmanbeg gazed out over the Sava plain.” The reader may know someone who is named after the river that runs through that plain.
The requirement to write down the text before recording was an official protocol of the Vienna Phonogrammarchiv, from which he had borrowed his equipment. Without this insistence on metadata, and thus the data-metadata conflict, I suppose what happened next might have had to wait several more decades –
Because this was the awful price of not seeing Homer as crazy: to see the Iliad and the Odyssey as folk art. Various scholars realized around 1900–1930 that the peculiarities of the Homeric poems were explained perfectly by seeing them as records of a living oral tradition. The repeated images were formulas that the singer, in the moment of singing, knew would bridge a gap in the meter or narrative. The poet had not memorized the text of the poem (because there was no such thing), only its vital shapes and a huge collection of words and techniques for recreating at the pace of speech.
Ascribing the epics to a particular man was natural for later people who saw writing as the natural form of language, and sole authorship as the natural source of literature, but these were collaborative works. Perhaps there was a Homer: maybe a man who, blind, was unable to learn reading and writing when his generation did, and spent his old age as the only one who remembered the songs they used to sing. Perhaps he started as a metaphor, or a just-so story for children, that got taken seriously. And maybe the Homeric poems were transcribed from a single telling by one bard, or maybe they were assembled like the bible out of various versions.
The Homeric and Balkan epics were the introduction to these ideas for contemporary Western scholarship, but there are hundreds of traditions like theirs all over the world. In fact, they are more or less the norm, or were, wherever writing was not common. We find them recorded in the world of writing, pinned butterflies, stuck in times and places when literacy was expanding. In my sketches for this Contextmas decoration, I was going to quote Robert Bringhurst’s translations and interpretations of Haida epics. It was his comparison of myths to fractals that got me started on this line of thought. But I had trouble narrowing the parts I wanted to discuss to less than several thousand words, so I will simply say that I owe a lot to him (here and elsewhere), and recommend all three books to anyone interested in literature. In the eastern United States, for example – a lot of what we know about certain oral traditions among slaves comes from right at their end, when they overlapped with writing for a single generation. There was a song that newly ex-slaves composed and recomposed about ambivalence toward death in a time of nominal freedom but great suffering, for instance, that was recorded in several quite different versions and we now know as Michael, Row the Boat Ashore. (This song is hobbyhorse of mine. I was made to sing it many times as a child and it never made sense. After I looked it up and considered the lyrics, I thought less of the adults who had presented it as a ditty.) A few generations later, another wave of literacy came through the same general area with the upheaval of the Great Depression and the electrification projects; Alan Lomax, for example, caught some hairs of another oral tradition’s tail on tape, where they inspired a lot of early white rock and roll.
People complain about the domestication of modern life – the mediation, the normality, the lack of invention. We are missing some wildness, some unmediated sensitivity and uninsulated toughness, that we need. We don’t fit our patterns to our fabric. This gets blamed on suburbs. I hope so. If it’s cars and the urban planning they enable, we can build trains and fix it. But in darker moments I think the problem might be the idea that every story has a definitive expression. This comes from recording technology – movies, albums, and especially writing. I hope literacy is not the problem, because I wouldn’t know how to take sides. I would have to think about everything case by case.
If you are a poet in an oral tradition, how do you actually compose/remember your poems? What is performance like?
For a really good answer, you should probably do something I haven’t done: ask someone who raps. Failing that, ask someone who plays jazz or other folk tunes.
A fellow I know who’s studied Beowulf – an oral epic – put it to me very neatly in a hurried e-mail:
A way I’ve described [Milman] Parry’s theory of oral formulaic poetic composition is something like this:
In regular speech, we have phonemes which are put together into words and we all agree on our collective Spot the etymological figure! collection of words. We also have a limited number of catchphrases and standard expressions, old sayings, proverbs, etc., but, for the most part, in regular speech, the largest unit which gets shifted around unchanged is the word.
In oral formulaic composition, there are larger units, phrases and sentences which get shifted around unchanged or mostly modestly changed. The oral poet constructs [their] utterance at the level of the formula rather than of the word. As well, an oral formulaic tradition will have a collection of standard scenes (the Arming of the Warrior is a good old Greek one picked up in the credits of Xena: Warrior Princess, BTW) which the poet can draw on.
Something that I remember getting my head around back in my studies was the fact that an oral formulaic tradition is not a system which helps a poet memorize a standard poem. Rather, it is a mechanism by which a poet can construct extempore a long narrative in verse, a narrative which remains the familiar story although being on the whole something never produced in quite the same way before.
I’m typing this on the evening of 29 December, 2012. I’ve just moved and I don’t have an internet connection at home except on my phone. I’m kind of enjoying it. Now I’m going to retell, with some differences to fit this context, something that I’ve said before but of which the only copy is online. I will neither try to say a new thing nor to match my old phrasing.
A hero on a beach
One of the most studied of these formulas is the Hero on the Beach, in the Anglo-Saxon tradition. It is a moment in which the protagonist stands on a beach, surrounded by companions, in the presence of a light, and begins a journey. Scholars found this in Beowulf and elsewhere, further and further afield, in increasingly subtle ways.
This went on for a decade or two and then a little paper, barely 2000 words, appeared in Neophilologus. Its author was a John He’s the fellow I quote above. Richardson, and instead of a university his affiliation was listed as Sherwood Park, Alberta – a hamlet near Edmonton. He asked, without malice, whether it was useful to go around spotting things that could be counted as the theme. He pointed out that, for example, Shakespeare’s version of Hamlet contains the elements of the theme if you try to find it there:
In scene IV of Act I, as Hamlet stands on the wall of Elsinore (an obvious border situation paralleled by an example in Guthlac) with his companions (retainers), his father’s ghostrevisits… the glimpses of the moonin full and presumably polished armor (both moon and armor fill the role of the light) and so begins Hamlet’s journey of self-discovery, revenge, and death.
But no one is prepared to argue that this is actually the theme, passed down continuously Though we do find a hint of oral-formulaic composition in Shakespare in a different way: I was very interested in the claim by Jonathon Hendry, at 21:45 to 42:45 in this Radio New Zealand Ideas interview, that Shakespeare tended to write his lines in ways that were easy to memorize – and, presumably, if the different versions of his plays that contemporary actors gave were an indication, to vamp and ad-lib without missing plot points. as a formula from Anglo-Saxon verse to early modern theater. Richardson’s argument as I read it is about pseudo-science and the criteron of falsifiability: if you don’t have a way of knowing when you’re wrong to apply your model or theory, you’re not learning much when it does seem to work.
If it isn’t the theme in Hamlet, what is it? It could be a coincidence. It could also be, as Richardson pointed out, an instance of a more general theme: a person in a liminal state, called on a quest. Joseph Campbell called this the crossing of the threshold. Not a verbal formula, then, but a clear metaphor for something that people recognize simply by virtue of being conscious, feeling, social things.
It may be that verbal formulas last if they align with these deeper emotional formulas. We don’t have a way of expressing feeling itself, but we can use our various tools, language among them, to draw progressively tighter and more detailed bands of color around the infinite edge of what we feel inside us, and know is in others, but doesn’t have its own external form. It might be clearest to say that the hero on the beach is an Anglo-Saxon oral formula, and also an image that comes a little more naturally to humans in general than one you would invent by throwing darts at a dictionary.
The poet Gary Snyder did his undergrad thesis on a Haida story called He Who Hunted Birds in His Father’s Village, which various people had pointed out was startlingly parallel to the swan-maiden folktale of Central Europe. In fact these both belong to a category of
animal wife stories that vary by region – seals in Northern Europe, cranes in East Asia, buffalo in Central Africa, and so on. I assume that if we could say what these stories mean, if it were as simple as
young men coming to terms with the otherness of women or something, we would already have spelled it out and wouldn’t need all the different stories.
My favorite instance of the Hero on the Beach is, by most standards other than historical ones, a really clear example: a protagonist (Carl Sagan) stands on a beach (a beach), surrounded by his companions (matter, energy, life, and the audience), in the presence of light (the sun glittering on the waves), and begins a journey (Cosmos):
- It’s similar at every scale;
- it’s full of continuous variations; and, conjecturally,
- it’s all connected.